Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels
Abstract Let K(x1,…,xn) $K(x_{1},\ldots,x_{n})$ satisfy K(x1,…,txi,…,xn)=tλλiK(t−λiλ1x1,…,xi,…,t−λiλnxn) $$K(x_{1},\ldots,tx_{i},\ldots,x_{n})=t^{\lambda\lambda _{i}}K \bigl(t^{-\frac{\lambda_{i}}{\lambda_{1}}}x_{1},\ldots,x_{i}, \ldots,t^{-\frac {\lambda _{i}}{\lambda_{n}}}x_{n} \bigr) $$ for t>...
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doaj-57be190c6c714a2ca74e92ddab2db2972020-11-24T21:51:49ZengSpringerOpenJournal of Inequalities and Applications1029-242X2018-08-012018111210.1186/s13660-018-1797-5Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernelsJunfei Cao0Bing He1Yong Hong2Bicheng Yang3Department of Mathematics, Guangdong University of EducationDepartment of Mathematics, Guangdong University of EducationCollege of Mathematics and Statistics, Guangdong University of Finance and EconomicsDepartment of Mathematics, Guangdong University of EducationAbstract Let K(x1,…,xn) $K(x_{1},\ldots,x_{n})$ satisfy K(x1,…,txi,…,xn)=tλλiK(t−λiλ1x1,…,xi,…,t−λiλnxn) $$K(x_{1},\ldots,tx_{i},\ldots,x_{n})=t^{\lambda\lambda _{i}}K \bigl(t^{-\frac{\lambda_{i}}{\lambda_{1}}}x_{1},\ldots,x_{i}, \ldots,t^{-\frac {\lambda _{i}}{\lambda_{n}}}x_{n} \bigr) $$ for t>0 $t>0$. With this integral kernel, by using the method and technique of weight coefficients, the equivalent conditions and the best constant factors for the validity of Hilbert-type integral inequalities involving multiple functions are discussed. Finally, the applications of the integral inequalities are considered.http://link.springer.com/article/10.1186/s13660-018-1797-5Hilbert-type integral inequalityQuasi-homogeneous kernelEquivalent conditionsBest constant factor |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Junfei Cao Bing He Yong Hong Bicheng Yang |
spellingShingle |
Junfei Cao Bing He Yong Hong Bicheng Yang Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels Journal of Inequalities and Applications Hilbert-type integral inequality Quasi-homogeneous kernel Equivalent conditions Best constant factor |
author_facet |
Junfei Cao Bing He Yong Hong Bicheng Yang |
author_sort |
Junfei Cao |
title |
Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels |
title_short |
Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels |
title_full |
Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels |
title_fullStr |
Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels |
title_full_unstemmed |
Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels |
title_sort |
equivalent conditions and applications of a class of hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels |
publisher |
SpringerOpen |
series |
Journal of Inequalities and Applications |
issn |
1029-242X |
publishDate |
2018-08-01 |
description |
Abstract Let K(x1,…,xn) $K(x_{1},\ldots,x_{n})$ satisfy K(x1,…,txi,…,xn)=tλλiK(t−λiλ1x1,…,xi,…,t−λiλnxn) $$K(x_{1},\ldots,tx_{i},\ldots,x_{n})=t^{\lambda\lambda _{i}}K \bigl(t^{-\frac{\lambda_{i}}{\lambda_{1}}}x_{1},\ldots,x_{i}, \ldots,t^{-\frac {\lambda _{i}}{\lambda_{n}}}x_{n} \bigr) $$ for t>0 $t>0$. With this integral kernel, by using the method and technique of weight coefficients, the equivalent conditions and the best constant factors for the validity of Hilbert-type integral inequalities involving multiple functions are discussed. Finally, the applications of the integral inequalities are considered. |
topic |
Hilbert-type integral inequality Quasi-homogeneous kernel Equivalent conditions Best constant factor |
url |
http://link.springer.com/article/10.1186/s13660-018-1797-5 |
work_keys_str_mv |
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1725878354206261248 |